L(s) = 1 | + 2·2-s − 8.51·3-s + 4·4-s + 16.7·5-s − 17.0·6-s + 25.1·7-s + 8·8-s + 45.5·9-s + 33.5·10-s − 16.1·11-s − 34.0·12-s + 62.6·13-s + 50.3·14-s − 143.·15-s + 16·16-s + 16.4·17-s + 91.0·18-s + 58.1·19-s + 67.1·20-s − 214.·21-s − 32.3·22-s − 170.·23-s − 68.1·24-s + 156.·25-s + 125.·26-s − 157.·27-s + 100.·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.63·3-s + 0.5·4-s + 1.50·5-s − 1.15·6-s + 1.35·7-s + 0.353·8-s + 1.68·9-s + 1.06·10-s − 0.443·11-s − 0.819·12-s + 1.33·13-s + 0.961·14-s − 2.46·15-s + 0.250·16-s + 0.234·17-s + 1.19·18-s + 0.702·19-s + 0.750·20-s − 2.22·21-s − 0.313·22-s − 1.54·23-s − 0.579·24-s + 1.25·25-s + 0.944·26-s − 1.12·27-s + 0.679·28-s + ⋯ |
Λ(s)=(=(538s/2ΓC(s)L(s)Λ(4−s)
Λ(s)=(=(538s/2ΓC(s+3/2)L(s)Λ(1−s)
Particular Values
L(2) |
≈ |
2.975533273 |
L(21) |
≈ |
2.975533273 |
L(25) |
|
not available |
L(1) |
|
not available |
L(s)=p∏Fp(p−s)−1 | p | Fp(T) |
---|
bad | 2 | 1−2T |
| 269 | 1−269T |
good | 3 | 1+8.51T+27T2 |
| 5 | 1−16.7T+125T2 |
| 7 | 1−25.1T+343T2 |
| 11 | 1+16.1T+1.33e3T2 |
| 13 | 1−62.6T+2.19e3T2 |
| 17 | 1−16.4T+4.91e3T2 |
| 19 | 1−58.1T+6.85e3T2 |
| 23 | 1+170.T+1.21e4T2 |
| 29 | 1+7.21T+2.43e4T2 |
| 31 | 1+126.T+2.97e4T2 |
| 37 | 1+188.T+5.06e4T2 |
| 41 | 1−15.2T+6.89e4T2 |
| 43 | 1+172.T+7.95e4T2 |
| 47 | 1−608.T+1.03e5T2 |
| 53 | 1−545.T+1.48e5T2 |
| 59 | 1−487.T+2.05e5T2 |
| 61 | 1−576.T+2.26e5T2 |
| 67 | 1+930.T+3.00e5T2 |
| 71 | 1−287.T+3.57e5T2 |
| 73 | 1−331.T+3.89e5T2 |
| 79 | 1−231.T+4.93e5T2 |
| 83 | 1+671.T+5.71e5T2 |
| 89 | 1+927.T+7.04e5T2 |
| 97 | 1+128.T+9.12e5T2 |
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L(s)=p∏ j=1∏2(1−αj,pp−s)−1
Imaginary part of the first few zeros on the critical line
−10.57332085386726448489013262844, −10.06599245586433303551269540845, −8.622723638975118981391460182528, −7.36067166543168933792350378491, −6.26380355694184789909730645779, −5.55160673363993102881892139392, −5.27656549813981684901962382098, −4.03743838081287627355022616560, −2.04362751581036137811721364359, −1.15123362350823851390247428078,
1.15123362350823851390247428078, 2.04362751581036137811721364359, 4.03743838081287627355022616560, 5.27656549813981684901962382098, 5.55160673363993102881892139392, 6.26380355694184789909730645779, 7.36067166543168933792350378491, 8.622723638975118981391460182528, 10.06599245586433303551269540845, 10.57332085386726448489013262844